Random path quantization

M. Roncadelli

pp. 407-416

in: Claudio Garola, Arcangelo Rossi (eds), The foundations of quantum mechanics, Berlin, Springer, 1995

Abstract

We offer a concise account of the Random Path Quantization (RPQ), whose motivation comes from the fact that quantum amplitudes satisfy (almost) the same calculus that probabilities obey in the theory of classical stochastic diffusion processes. Indeed — as a consequence of this structural analogy — a new approach to quantum mechanics naturally emerges as the quantum counterpart of the Langevin description of classical stochastic diffusion processes: This is just the RPQ. Starting point is classical mechanics as formulated a là Hamilton-Jacobi. Quantum fluctuations enter the game through a certain white noise added in the first-order equation that yields the configuration space trajectories as controlled by the solutions of the (classical) Hamilton-Jacobi equation. A Langevin equation arises in this way and provides the quantum random paths. The quantum mechanical propagator is finally given by a noise average involving the quantum random paths (in complete analogy with what happens for the transition probability of a classical stochastic diffusion process within the conventional Langevin treatment). The general structure of the RPQ is discussed, along with a suggested intuitive picture of the quantum theory.